Hurewicz theorem

In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism. The theorem is named after Witold Hurewicz, and generalizes earlier results of Henri Poincaré. The Hurewicz theorems are a key link between homotopy groups and homology groups. For any path-connected space X and positive integer n there exists a group homomorphism called the Hurewicz homomorphism, from the n-th homotopy group to the n-th homology group (with integer coefficients). It is given in the following way: choose a canonical generator , then a homotopy class of maps is taken to . The Hurewicz theorem states cases in which the Hurewicz homomorphism is an isomorphism. For , if X is -connected (that is: for all ), then for all , and the Hurewicz map is an isomorphism. This implies, in particular, that the homological connectivity equals the homotopical connectivity when the latter is at least 1. In addition, the Hurewicz map is an epimorphism in this case. For , the Hurewicz homomorphism induces an isomorphism , between the abelianization of the first homotopy group (the fundamental group) and the first homology group. For any pair of spaces and integer there exists a homomorphism from relative homotopy groups to relative homology groups. The Relative Hurewicz Theorem states that if both and are connected and the pair is -connected then for and is obtained from by factoring out the action of . This is proved in, for example, by induction, proving in turn the absolute version and the Homotopy Addition Lemma. This relative Hurewicz theorem is reformulated by as a statement about the morphism where denotes the cone of . This statement is a special case of a homotopical excision theorem, involving induced modules for (crossed modules if ), which itself is deduced from a higher homotopy van Kampen theorem for relative homotopy groups, whose proof requires development of techniques of a cubical higher homotopy groupoid of a filtered space.